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Randomness, lowness and degrees. (English) Zbl 1145.03020

Summary: We say that \(A\leq _{LR}B\) if every \(B\)-random number is \(A\)-random. Intuitively this means that if oracle \(A\) can identify some patterns on some real \(\gamma \), oracle \(B\) can also find patterns on \(\gamma \). In other words, \(B\) is at least as good as \(A\) for this purpose. We study the structure of the \(LR\) degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever \(\alpha \) is not \(GL_{2}\) the \(LR\) degree of \(\alpha \) bounds \(2^{\aleph _{0}}\) degrees (so that, in particular, there exist \(LR\) degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. \(LR\) degrees.

MSC:

03D30 Other degrees and reducibilities in computability and recursion theory
03D25 Recursively (computably) enumerable sets and degrees
03D28 Other Turing degree structures
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